Absolute Value

Absolute Value Problem Solving


Let x=70x=70 be one root of the equation 2x10=x+a,2\lvert x-10\rvert=x+a, where aa is a constant. Then what is the other root?

Let SS be the solution set of the equation 2x1+3x29=5x88.\lvert 2x-1 \rvert + 3\lvert x-29\rvert=5x-88. Then what is S?S?

How many ordered pairs (x,y)(x, y) are there such that xx is a nonzero integer that satisfies 8x8-8\leq x \leq 8, yy is a nonzero integer that satisfies 2y2-2\leq y \leq 2, and x+y=x+y?\lvert x+y \rvert = \lvert x \rvert + \lvert y \rvert?

Details and assumptions

For an ordered pair of integers (a,b)(a,b), the order of the integers matter. The ordered pair (1,2)(1, 2) is different from the ordered pair (2,1)(2,1) .

How many ordered triples of integers (a,b,c) (a, b, c) subject to 20<a<b<c<20, -20 < a < b < c < 20, are there, such that

a+b+c=a+b+c? \left| a + b + c \right | = |a| + |b| + |c|?

If 20<a2120 < a \leq 21, what is the value of 2a6a20? \bigg| 2-a- \big| 6- \lvert a-20 \rvert \big| \bigg| ?


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